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Theorem rexlimddvcbv 44220
Description: Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 44218. The equivalent of this theorem without the bound variable change is rexlimddv 3161. Usage of this theorem is discouraged because it depends on ax-13 2377, see rexlimddvcbvw 44219 for a weaker version that does not require it. (Contributed by Rohan Ridenour, 3-Aug-2023.) (New usage is discouraged.)
Hypotheses
Ref Expression
rexlimddvcbv.1 (𝜑 → ∃𝑥𝐴 𝜃)
rexlimddvcbv.2 ((𝜑 ∧ (𝑦𝐴𝜒)) → 𝜓)
rexlimddvcbv.3 (𝑥 = 𝑦 → (𝜃𝜒))
Assertion
Ref Expression
rexlimddvcbv (𝜑𝜓)
Distinct variable groups:   𝜑,𝑦   𝜓,𝑦   𝜒,𝑥   𝜃,𝑦   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑥)

Proof of Theorem rexlimddvcbv
StepHypRef Expression
1 rexlimddvcbv.1 . 2 (𝜑 → ∃𝑥𝐴 𝜃)
2 rexlimddvcbv.3 . . 3 (𝑥 = 𝑦 → (𝜃𝜒))
3 rexlimddvcbv.2 . . 3 ((𝜑 ∧ (𝑦𝐴𝜒)) → 𝜓)
42, 3rexlimdvaacbv 44218 . 2 (𝜑 → (∃𝑥𝐴 𝜃𝜓))
51, 4mpd 15 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2108  wrex 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2377
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071
This theorem is referenced by: (None)
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